From Kernel Matrices to Kernel Functions: An Eigenfunction-Based Approach

Kernel-combination procedures used in classification often return only a combined kernel matrix on the training sample, rather than a kernel function that can be evaluated consistently at new points. This limitation is especially important for supervised or label-aware combinations, whose entries may depend on training labels and therefore have no immediate out-of-sample meaning. We study the problem of constructing an inductive, finite-rank kernel extension from such empirical matrices. The proposed framework makes the non-uniqueness of this extension explicit: it is determined by empirical coordinates, a positive-semidefinite coefficient matrix, and a continuation model for the coordinates. Experiments on vector, tabular, and relational classification problems give a deliberately diagnostic picture. Smooth direct combinations are stable: on Synthetic, the direct mean gives error $0.0793\pm 0.0227$, essentially matching the best individual RBF kernel ($0.0809\pm 0.0231$), and on Telco it remains close to the best individual polynomial kernel ($0.2061\pm 0.0154$ versus $0.2045\pm 0.0154$). In the controlled Synthetic oracle diagnostic, reconstructing a smooth sum/mean gives relative Frobenius error $4.13\times {10}^{-6}\pm 9.41\times {10}^{-6}$ and functional MSE at numerical scale. By contrast, abrupt label-aware matrix-only rules are less robust: the Synthetic percentile_inout_auto rule has error $0.1404\pm 0.1198$, Telco matrix-only supervised rules are around $0.307$–$0.326$ error, and the Chickenpieces pickout_auto rule fails under strict out-of-sample reconstruction ($0.3545\pm 0.2666$ error), whereas direct relational combinations match the best individual relational kernel within ${10}^{-3}$. Overall, the empirical evidence supports the method as a bridge from finite matrix-level information fusion to deployable kernels, while also identifying abrupt label-aware geometries as the main limitation for stable generalization.